Welcome to Modelling Extra Risk!
Hello there! As you dive into the world of long-term insurance, you’ll quickly realize that not every person fits the "standard" mold. Some people smoke, some have high blood pressure, and others might have high-risk hobbies like skydiving. In actuarial terms, we call these substandard risks.
In this chapter, we are going to learn how to adjust our mathematical models to account for this extra risk. Don't worry if this seems tricky at first—we’re basically just taking the "standard" formulas you already know and giving them a little "tweak" to make them fit more dangerous situations. Let's get started!
Why Do We Need to Adjust Mortality?
Standard mortality tables are built for average, healthy people. If we used those same tables for someone with a chronic illness, we would underprice the insurance. The company wouldn't collect enough premium to cover the higher chance of a claim. To fix this, we use extra risk models to increase the mortality rates used in our calculations.
Method 1: Age Rating (The "Odometer" Method)
Age rating (also known as "adding to the age") is one of the simplest ways to handle extra risk. The idea is simple: if a 40-year-old applicant smokes, their body might "act" like the body of a 45-year-old.
How it Works
We simply add a fixed number of years, let’s call it \(h\), to the applicant’s actual age. If the true age is \(x\), we perform all our calculations as if the person is aged \(x + h\).
For example, if a 35-year-old is given a 5-year age rating:
- The death probability becomes: \(q_{35}^* = q_{40}\)
- The life annuity becomes: \(\ddot{a}_{35}^* = \ddot{a}_{40}\)
- The net premium becomes: \(P_{35}^* = P_{40}\)
An Analogy
Think of it like a car. A 2020 model car that was used for heavy deliveries might have 150,000 miles on it. Even though it's technically only 4 years old, an insurance company would treat it like a much older car because of the "wear and tear." That is exactly what age rating does to a human life!
Quick Review: With Age Rating, just replace \(x\) with \(x+h\) in every formula you know. It’s that easy!
Method 2: Constant Additions to the Force of Mortality
Sometimes, the extra risk isn't about being "older." Sometimes, there is just a flat, extra layer of risk that stays the same every year. This is modeled by adding a constant, \(\phi\) (phi), to the force of mortality.
The Formula
The adjusted force of mortality, \(\mu_x^*\), is defined as:
\( \mu_{x+t}^* = \mu_{x+t} + \phi \)
What does this do to our survival probability?
This is a favorite topic for FAM exam questions! Because the force of mortality is in the exponent of the survival function, adding a constant results in a multiplicative adjustment to the survival probability:
\( {}_tp_x^* = {}_tp_x \cdot e^{-\phi t} \)
Did you know?
This method is often used for "flat" risks, like a person who has a dangerous hobby. Whether they are 20 or 50, the extra risk of a skydiving accident might be considered relatively constant every year.
Key Takeaway: If you see a constant addition \(\phi\), remember that your new survival probability is just the old one multiplied by \(e^{-\phi t}\). This makes calculating \(A_x^*\) or \(\ddot{a}_x^*\) much faster!
Method 3: Multiplicative Adjustments (The Percentage Method)
In this method, we assume the extra risk is proportional to the standard risk. We multiply the standard force of mortality by a factor, usually written as \((1+k)\) or a constant \(m\).
The Formula
\( \mu_x^* = (1+k) \cdot \mu_x \)
Common Pitfall
Students often mistake this for multiplying the probability of death (\(q_x\)). Be careful! In this model, we are multiplying the force of mortality (\(\mu\)). While they are related, multiplying the force has a compounding effect over time.
Under this model, the new survival probability is:
\( {}_tp_x^* = ({}_tp_x)^{(1+k)} \)
How These Adjustments Affect Premiums and Values
Once you have adjusted the mortality (either by age rating or changing \(\mu\)), you must recalculate the Actuarial Present Values (APV) to find the new premiums.
Step-by-Step Calculation Process:
- Identify the adjustment: Is it age rating? Is it a constant \(\phi\)? Or a multiplier?
- Calculate the adjusted survival probabilities: Find the new \({}_tp_x^*\).
- Calculate the adjusted APVs: Find the new \(A_x^*\) and \(\ddot{a}_x^*\).
- Find the Premium: Use the Equivalence Principle. Usually, \(P^* = \frac{A_x^*}{\ddot{a}_x^*}\).
Important Point: When mortality increases, \(A_x\) (the cost of the benefit) increases, and \(\ddot{a}_x\) (the value of the premiums) decreases. Because the numerator goes up and the denominator goes down, the premium \(P_x\) will always be higher for a substandard risk.
Summary Table for Quick Reference
Use this table to keep the different methods straight in your head!
1. Age Rating: Treat age \(x\) as \(x+h\).
2. Constant Addition (\(\phi\)): \({}_tp_x^* = {}_tp_x \cdot e^{-\phi t}\).
3. Multiplicative (\(m\)): \({}_tp_x^* = ({}_tp_x)^m\).
Final Tips for the Exam
- Watch the timing: If a question mentions "extra risk applies only for the first 5 years," only adjust the mortality for those 5 years. This is a common "trick" on the FAM exam!
- Policy Values: If you are asked for the policy value (\(_tV\)) of a substandard life, remember that both the future benefits and future premiums are based on the adjusted mortality.
- Sanity Check: If your calculation results in a premium that is lower than the standard premium, stop! Something went wrong. Extra risk should always lead to higher costs.
You're doing great! This chapter is all about understanding how small changes to our assumptions flow through the standard formulas. Master the three adjustment methods above, and you'll be ready for any "extra risk" question the SOA throws at you!